Acoustic phonon scattering limited carrier mobility in 2D extrinsic graphene
We theoretically calculate the phonon scattering limited electron mobility in extrinsic (i.e. gated or doped with a tunable and finite carrier density) 2D graphene layers as a function of temperature and carrier density . We find a temperature dependent phonon-limited resistivity to be linear in temperature for with the room temperature intrinsic mobility reaching values above cm. We comment on the low-temperature Bloch-Grüneisen behavior where for unscreened electron-phonon coupling.
pacs:81.05.Uw, 72.10.-d, 73.40.-c
Low temperature carrier transport properties of 2D graphene layers have been of great current interest to both experimentalists Novoselov et al. (2004); Zhang et al. (2005); Tan (); Fuhrer (); review () and theorists Ando (); Nomura (); Cheianov and Fal’ko (2006); Hwang1 (); Adam () alike ever since the possibility of fabricating stable gated 2D graphene monolayers on SiO substrates and measuring the density-dependent conductivity of the 2D chiral graphene carriers were demonstrated Novoselov et al. (2004). Much of the early interest focused understandably on the important issues of the scattering mechanisms limiting the low-temperature conductivity and the associated graphene “minimal conductivity” at the charge neutral (“Dirac”) point. One of the dominant low-temperature scattering mechanisms Tan (); Fuhrer (); Hwang1 (); Adam () in graphene is that due to screened Coulomb scattering by unintended charged impurities invariably present in the (mostly SiO) substrate (and the substrate-graphene interface) although short-range scattering by neutral defects also contributes, particularly at high carrier densities in high-mobility samples. It has therefore been argued Hwang1 (); Adam () that gated graphene layers are similar to 2D electron systems in confined semiconductor structures (e.g. Si inversion layers, GaAs heterostructures and quantum wells) where also long-range charged impurity scattering dominates low-temperature ohmic transport with short-range (e.g. interface roughness) scattering playing a role at high carrier densities. AndoRMP (); Manfra ()
Given the exciting technological context of graphene as a prospective electronic transistor material for future applications, the question therefore naturally arises about the limiting value of the intrinsic room-temperature graphene mobility if all extrinsic scattering mechanisms, e.g. charged impurities, neutral defects, interface roughness, graphene ripples etc., can be eliminated from the system. This question is more than of academic interest since serious experimental efforts are underway Stomer () to eliminate charged impurities from graphene by using different substrates or by working with free standing graphene layers without any substrates. It is also noteworthy that the systematic elimination of charged impurity scattering through modulation doping and materials improvement in the MBE growth technique has led to an astonishing 3,000-fold enhancement in the low-temperature 2D GaAs electron mobility from cm in 1978 to cm in 2000, future enhancement to 100 million cm mobility is anticipated Pfeiffer () in the next few years.
One great advantage of graphene over high-mobility 2D GaAs systems is that the lack of strong long-range polar optical phonon scattering, which completely dominates Kawamura1 () the room-temperature GaAs mobility ( cm), in graphene should lead to very high intrinsic room temperature graphene mobility, limited only by the weak deformation-potential scattering from the thermal lattice acoustic phonons. In this work, we calculate the temperature-dependent 2D graphene mobility limited only by the background lattice acoustic phonon scattering. We find that room-temperature intrinsic (i.e. just phonon-limited) graphene mobility surpassing cm is feasible using the generally accepted values in the literature for the graphene sound velocity and deformation coupling. There is some uncertainty in the precise value of the electron-phonon deformation potential coupling constant, leading to a concomitant uncertainty in the intrinsic graphene mobility. This situation is similar Kawamura2 () to the 2D GaAs system where in fact precise measurement of the phonon-limited 2D mobility led to the correct deformation potential coupling for transport studies, and this could be the case for graphene also where a quantitative comparison between our theoretical results presented in this work with the measured temperature-dependent graphene mobility, very recently becoming available Fuhrer2 (); Geim2 (), could lead to an accurate determination of the graphene electron-phonon deformation potential coupling constant.
The paper is organized as follows. In section II the Boltzmann transport theory is presented to calculate acoustic phonon scattering limited 2D graphene conductivity. Section III presents the results of the calculation. In section IV we discuss the results compared to experimental data, and we conclude in Section V.
We use the Boltzmann transport theory Kawamura1 (); Kawamura2 () to calculate acoustic phonon scattering limited 2D graphene conductivity. We consider only the longitudinal acoustic (LA) phonons in our theory since either the couplings to other graphene lattice phonon modes are too weak or the energy scales of these (optical) phonon modes are far too high for them to provide an effective scattering channel in the temperature range () of our interest.
The conductivity of graphene is given by
where is the Fermi velocity, is the density of states of graphene at the Fermi level () and is the relaxation time averaged over energy, i.e.
where is the Fermi distribution function, with and as the finite temperature chemical potential determined self-consistently. The energy dependent relaxation time  is defined by
where is the scattering angle between and , , and is the transition probability from the state with momentum to state. In this paper we only consider the relaxation time due to deformation potential (DP) coupled acoustic phonon mode. The deformation potential due to quasi-static deformation of lattice is taken into account. Then the transition probability has the form
where is the matrix element for scattering by acoustic phonon and is given by
where is the acoustic phonon energy with being the phonon velocity and is the phonon occupation number
The first (second) term is Eq. (5) corresponds to the absorption (emission) of an acoustic phonon of wave vector . The matrix element is independent of the phonon occupation numbers. The matrix element for the deformation potential is given by
where is the deformation potential coupling constant, is the graphene mass density, and is the area of the sample.
The scattering of electrons by acoustic phonons may be considered quasi-elastic since , where is the Fermi energy. There are two transport regimes, which apply to the temperature regimes and , depending on whether the phonon system is degenerate (Bloch-Grüneisen, BG) or non-degenerate (equipartition, EP). The characteristic temperature is defined as , which is given, in graphene, by K with density measured in unit of . First we consider . In this case we have , and . Then the relaxation time is calculated to be
Thus, in the non-degenerate EP regime () the scattering rate  depends linearly on the temperature. Since at low temperatures () the calculated conductivity is independent of Fermi energy or equivalently the electron density. Therefore the electronic mobility in graphene is inversely proportional to the carrier density, i.e. . The EP regime has recently been considered in the literature Geim2 (); Stauber (). We note that the similar linear temperature dependence of the scattering time has been reported for nanotubes Kane () and graphites phonon ().
To calculate the relaxation times in the BG regime where we have to keep the full form as in Eq. (5). Since the acoustic-phonon energy is comparable to the temperature dependence of the relaxation time via the statistical occupation factors in Eq. (5) becomes more complicated. In BG regime the scattering rate is strongly reduced by the occupation factors because for phonon absorption the phonon population decreases exponentially and also phonon emission is prohibited by a sharp Fermi distribution. To calculate the low temperature behavior of the resistivity we can rewrite the averaged inverse scattering time over energy as
where and is given by Price ()
Then we have in low temperature limits
Thus, we find that the temperature dependent resistivity in BG regime becomes without screening effects. If we include screening effects by the carriers themselves Hwang2 () the low-temperature resistivity goes as . The screening effects on the bare scattering rates can be introduced by dividing the matrix elements by the dielectric function of graphene. But the matrix elements in graphene arise from the change in the overlap between orbitals placed on different atoms and not from a Coulomb potential. Thus, we neglect screening effects in the calculation, and only consider unscreened deformation potential coupling. Even though the resistivity in EP regime is density independent, Eq. (11) indicates that the calculated resistivity in BG regime is inversely proportional to the density, i.e. , or equivalently the mobility in BG regime is density independent.
In this calculation we use the following parameters: graphene mass density , acoustic phonon velocity , and deformation potential . Even though the phonon velocity is well defined experimentally the value of the deformation potential coupling constant is not established Deformation (); phonon (). In general, the constant could be obtained on the basis of the fact that the shift of energy dispersion from its equilibrium state reaches the order of the atomic energy, i.e. with being the lattice constant, which is of the order of 10 eV in graphene. We note that we have used ref. Deformation, for obtaining the phonon parameters, but different values of , differing by factors of three (i.e. eV), are quoted Deformation (); phonon (); Stauber () in the literature. Since , the resulting graphene resistivity could differ by an order of magnitude depending on the precise value of .
In Fig. 1 we show the calculated inverse relaxation times for deformation potential scattering by acoustic phonon as a function of energy for different temperatures , 0.5, 1.0, and 1.5 for an electron density cm with . The inverse relaxation time in BG regime () shows a characteristic dip (suppression of scattering rate) in a narrow region around Fermi energy due to the statistical occupation factors. Above Bloch-Grüneisen temperature () the dip structure disappear and the scattering rate becomes close to the scattering rate of equipartition regime.
In Fig. 2(a) we show our calculated graphene resistivity, , as a function of temperature on a log-log plots, clearly demonstrating the two different regimes: BG behavior for and the EP behavior for . We note that () depends weakly on density, and the true BG behavior is likely to show up at relatively low temperatures. In Fig. 2(b), we show for several densities on a linear plot, emphasizing the strong linear in dependence of the acoustic phonon-limited graphene resistivity ranging from 100K to 500K. This rather large temperature range of behavior of acoustic phonon scattering limited resistivity is quite generic to 2D semiconductor structures, and our finding for graphene here is qualitatively similar to what was earlier found to be the ease for 2D GaAs structures Kawamura1 (); Kawamura2 (). The crucial difference between graphene and 2D GaAs is that in the latter system polar optical phonon scattering becomes exponentially more important for and dominates at room temperatures whereas in graphene we predict a linear 2D resistivity upto very high temperatures () since the relevant optical phonon have very high energy () and are simply irrelevant for carrier transport.
In Fig. 3, we show our calculated intrinsic graphene mobility, , as functions of temperature and carrier density. Within our model, the unscreened acoustic phonon scattering limited graphene mobility is inversely proportion to and individually for . Assuming , as used in Fig. 3, could reach values as high as cm for lower carrier densities ( cm). For larger (smaller) values of , would be smaller (larger) by a factor of . It may be important to emphasize here that we know of no other system where the intrinsic room-temperature carrier mobility could reach a value as high as cm.
The three key theoretical findings on phonon-limited graphene mobility of this work are : (1) for ; (2) () for ) for unscreened (screened) deformation potential coupling; (3) cm at room temperature () where is measured in eV and is carrier density measured in units of cm. These theoretical predictions being rather precise, the question naturally arises about the experimental status and the verification of our theory. Very recently, experimental graphene transport data at room temperature (or even above Fuhrer2 ()) have started becoming available Fuhrer2 (); Geim2 (). The only aspect of our theory that can be directly compared with the existing experiment is the behavior at high temperatures (), and this is indeed consistent with the recent data from two different groups Fuhrer2 (); Geim2 (). Geim and collaborators have recently concluded Geim2 () that the room temperature graphene intrinsic mobility could be as high as cm, which is also consistent with our theory. But, as emphasized by us, the actual mobility value varies inversely as , and therefore a precise knowledge of the deformation potential coupling is required for an accurate estimate of the intrinsic mobility.
A more detailed comparison between our theory and the experimental results on shows some qualitative difference which are not understood at this point. For example, the experimental crossover Fuhrer2 (); Geim2 () to the high-temperature linear () behavior in the intrinsic resistivity appears to be closer to a behavior rather than the BG behavior we predict. More disturbingly, the experimental crossover from the high-temperature linear behavior to the low-temperature high power law behavior appears to be occurring at a much higher temperature () than the theoretical prediction (). At this stage we have no explanation for the lower-temperature disagreement between experiment and theory, but below we discuss several possibilities.
The experimentally measured resistivity in the current graphene samples is completely dominated by extrinsic scattering (and not by phonon scattering) even at room temperatures since the low-temperature () mobility is typically cm, and the intrinsic room-temperature phonon contribution, as obtained theoretically by us or inferred Fuhrer2 (); Geim2 () from recent temperature-dependent experiments, is times larger (cm). This means that any experimental extraction of the pure phonon contribution to graphene resistivity involves subtraction of two large resistances (i.e. the measured total resistance and the extrapolated extrinsic temperature-independent resistance arising from impurity and defect scattering) of the order of each to get a phonon contribution roughly of the order of 100 . Apart from the inherent danger of large unknown errors involved in the subtraction of two large numbers to obtain a much smaller number associated with phonon scattering contribution to graphene mobility, there is the additional assumption of the Matthiessen’s rule, i.e. where is the total resistivity contributed by impurities and defects () and phonons (), which is simply not valid. In particular, the impurity contribution to resistivity also has a temperature dependence arising from Fermi statistics and screening which, although weak, cannot be neglected in extracting the phonon contribution (particularly since the total phonon contribution itself is much smaller than the total extrinsic contribution). In particular, the temperature dependent part of the charged impurity scattering contribution to graphene resistivity could be positive or negative Hwang3 () depending on whether screening or degeneracy effects dominate, and therefore the phonon contribution, as determined by a simple subtraction, could have large errors, particularly in the low () temperature regime. Indeed, a recent measurement Kim_T () of in the regime finds small temperature dependent contributions to graphene resistivity which could be either positive or negative depending on the sample mobility and which, in all likelihood, arises from extrinsic impurity scattering.
In Fig. 4 we show the failure of the Matthiessen’s rule in graphene (particularly at higher/lower temperatures/densities) by calculating the total graphene resistivity arising from screened charged impurity scattering () and phonon scattering () — it is clear that for lower/higher densities/temperatures.
If the experimentally extracted Fuhrer2 (); Geim2 () phonon contribution to the graphene resistivity turns out to be accurate in spite of the rather questionable subtraction procedure discussed above, then the disagreement between our intermediate temperature (50—100K) theoretical results and the experimental data would indicate the presence of some additional phonon modes which must be participating in the scattering process. We can, in fact, get reasonable agreement between our theory and the experimental data by arbitrarily shifting the BG temperature to a higher temperature around 200K. This shift could indicate a typical phonon scale which causes additional scattering other than the LA phonons coupled to the carriers through the deformation potential coupling considered in our work. At this stage we cannot speculate on what these additional modes could be. One possibility is that these are the zone-edge out of plane ZA phonon modes with vibrations transverse to the graphene plane Fuhrer2 (). (In the Appendix we provide the calculated carrier resistivity in the presence of an additional phonon mode with a soft gap.) Another possibility considered in ref. Geim2, is that these are the thermal fluctuations (”ripplons”) of the mechanical ripples invariably present in graphene samples Adam (). Of course such additional “phonon” scattering channels will lead to additional unknown coupling parameters making the resultant theory essentially a data fitting procedure. The advantage of our minimal theory is that it involves only two phonon parameters: and associated with the 2D graphene LA phonons. More data in higher mobility samples will be needed to settle this question since the subtraction problem inherent in the current technique for extracting the phonon contribution would make analyzing this issue a difficult task.
We have calculated the intrinsic temperature dependent 2D graphene transport behavior upto 500K by considering temperature and density dependent scattering of carriers by acoustic phonons. We have provided a critical discussion of our results in light of the recent experiments Fuhrer2 (); Geim2 (). The lack of precise quantitative knowledge about graphene deformation potential coupling makes a quantitative comparison with the experimental data problematic.
In this appendix we calculate the phonon scattering limited carrier mobility including effects of two phonon branches: the regular LA phonon (as considered in the main part of this paper) and an additional “intervalley” phonon branch with a soft gap ( meV) representing the inter-valley phonon, the ZA phonon mode at the K point. The theoretical motivation is to demonstrate that the combination of the LA phonon and an optical phonon (i.e. the inter-valley ZA mode) with a softy gap could indeed lead to qualitative (or even quantitative, if the phonon parameters have appropriate values) agreement between theory and experiment.
In this context we consider the long wavelength phonon scattering. As temperature increases phonons of large wave vectors are involved in the scattering in multi-valley structures. Thus the inter-valley phonon scattering becomes significant at high temperatures Fuhrer2 (). In graphene there are two minima of the conduction band at K and K’ points in Brillouin zone. The scattering between K and K’ points requires the participation of inter-valley phonons, whose wave vectors are close to q and the frequencies of these phonons are close to . The relaxation time for inter-valley phonon scattering may be considered by assuming constant inter-valley phonon energies . Then the matrix element for the inter-valley phonon scattering becomes where is the deformation potential coupling constant for inter-valley phonons in unit of eV/Å. Since in graphene the scattering of electrons from inter-valley phonons is considered quasielastically.
In Fig. 5 we show the calculated resistivity with both the acoustic phonon scattering and the inter-valley phonon scattering as a function of temperature for two different densities. The following parameters are used in this calculation: deformation potential coupling constant , acoustic phonon velocity cm/s, eV/Å, and inter-valley phonon energy meV which corresponds to the lowest phonon energy (ZA) at the K point. The calculated resistivities have very weak density dependence, which comes from the energy averaging. Below 200K the acoustic phonon scattering dominates (linear in temperature), but above 200 K both phonon scatterings contribute in the transport. Note that the temperature dependence of the resistivity is linear in both regimes, but has different slopes. The high power law behavior () only applies at very low temperatures ( K). Basically, there is a sharp turn-on in phonon scattering in the 150-250K range as the inter-valley ZA phonon scattering becomes effective.
Whether the results shown in Fig. 5, including the effects of inter-valley phonon scattering are physically meaningful or not will depend on the direct observation of these ZA phonon modes via Raman scattering experiments. At this stage, all we have established is that inclusion of this additional soft-gap phonon mode gives impressive agreement between theory and experiment Fuhrer2 (); Geim2 (). More work is needed to validate the model of combined LA and ZA phonon scattering contributing to the temperature dependent graphene resistivity.
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